Slack sensitivity to parameter variation based timing analysis

ABSTRACT

A method, system and program product are disclosed for improving an IC design that prioritize failure coefficients of slacks that lead to correction according to their probability of failure. With an identified set of independent parameters, a sensitivity analysis is performed on each parameter by noting the difference in timing, typically on endpoint slacks, when the parameter is varied. This step is repeated for every independent parameter. A failure coefficient is then calculated from the reference slack and the sensitivity of slack for each of the timing endpoints and a determination is made as to whether at least one timing endpoint fails a threshold test. Failing timing endpoints are then prioritized for modification according to their failure coefficients. The total number of runs required is one run that is used as a reference run, plus one additional run for each parameter.

BACKGROUND OF THE INVENTION

1. Technical Field

The present invention relates generally to integrated circuit design,and more particularly, to a method, system and program product forimproving IC design performance by analyzing IC timing based on a slacksensitivity to variations in parameters that affect timing.

2. Related Art

A wide variety of methods are employed in the optimization of integratedcircuit designs. One of these methods includes evaluation of the statictiming of parts of the circuit. Methodologies used today for statictiming analysis are based on different parameters that affect timing.The parameter values used are typically based on nominal values ofparameters or on extreme values of the range of values for a parameter,for example, a minimum and maximum value corresponding to ±3 standarddeviation units. In order to evaluate the impact that a given parameterwill have on timing, a timing run is performed with the parameter set toa minimum value, and then another timing run is performed with theparameter set to a maximum value.

As technology offerings become richer in terms of increased features(e.g., the number of devices offered, increased metal layers orincreased number of voltage islands), the number of parameters thataffect timing grows accordingly. For example, the number of parameterscan easily exceed 10 or more in today's technologies. Unfortunately,conventional techniques to evaluate timing over all parameters, requires2^(N) timing runs, where N represents the number of parameters. Each ofthe 2^(N) timing runs uses a different combination of minimum andmaximum settings for different parameters. Any one combination isreferred to herein as a “parameter/process permutation.” Where thenumber of parameters exceeds 10, timing analysis can require more than1000 timing runs for a complete evaluation, which is extremely time andresource consuming.

In addition to the above problem, conventional timing analysisapproaches also do not satisfactorily address timing errors because theytypically focus exclusively on timing endpoint (latch) slack, i.e., thedifference between a timing requirement and an actual timing caused bythe circuit. FIG. 1 shows one illustrative conventional approach. Inthis approach, in step S1, a timing run is conducted using a singleparameter/process permutation to determine endpoint slacks. These slacksare not necessarily bounding (i.e., extreme values) since other choicesfor parameter/process permutations could easily lead to moreconservative slacks. As a result, the conventional approach may yieldcircuit timing analysis results that fail to detect potential timingerrors and lead to non-zero failure probabilities.

In step S2, these non-bounding slacks are then compared to a slackthreshold. Slacks less than slack threshold S(threshold), whereS(threshold) is typically chosen as zero, are considered failures thatrequire correction. Unfortunately, timing endpoints with the largestnegative slacks may not represent the most critical slacks in terms ofmaximizing product yield because their sensitivities to parametervariations may result in a smaller probability of failure as compared toother timing endpoints. In other words, using a single parameter/processpermutation does not adequately address the sensitivities to parametervariations that may lead to failure. In addition, the conventionalapproach provides no insight relative to parameter non-trackingsensitivities, i.e., situations where two or more parameters that arenormally expected to vary together in a particular manner do not varytogether. Therefore, using slack values to prioritize timing endpointsfor correction can lead to poor utilization of resources.

In step S3, the failing endpoints and corresponding slack values aretargeted for correction. Since only the slack value from a timinganalysis at a single parameter/process permutation is available to guidecorrection, the correction cannot improve design robustness againstparameter variations or maximize chip performance under all variations.After correction, the conventional methodology returns to step S1 forvalidation.

In view of the foregoing, there is a need in the art for a way toaddress the problems of the related art.

SUMMARY OF THE INVENTION

The invention includes a method, system and program product forimproving an IC design that prioritize failure coefficients of slacksthat lead to correction according to their probability of failure. Withan identified set of independent parameters, a sensitivity analysis isperformed on each parameter by noting the difference in timing,typically on endpoint slacks, when the parameter is varied. This step isrepeated for every independent parameter. A failure coefficient is thencalculated from the reference slack and the sensitivity of slack foreach of the timing endpoints and a determination is made as to whetherat least one timing endpoint fails a threshold test. Failing timingendpoints are then prioritized for modification according to theirfailure coefficients. The total number of runs required is one run thatis used as a reference run, plus one additional run for each parameter.

A first aspect of the invention is directed to a method of improving aprobability of an integrated circuit (IC) design meeting timingrequirements, the method comprising the steps of: a) determining areference slack and a sensitivity of slack to a variation in at leastone parameter for each of a plurality of timing endpoints of the design;b) calculating a failure coefficient from the reference slack and thesensitivity of slack for each of the timing endpoints; c) determiningwhether each timing endpoint fails a threshold test; d) prioritizing anytiming endpoints that fail the threshold test according to respectivefailure coefficients; and e) modifying the design to improve a slack ofat least one of the timing endpoints.

A second aspect of the invention is directed to a system for improving aprobability of an integrated circuit (IC) design meeting timingrequirements, the system comprising: a) means for determining areference slack and a sensitivity of slack to a variation in at leastone parameter for each of a plurality of timing endpoints of the design;b) means for calculating a failure coefficient from the reference slackand the sensitivity of slack for each of the timing endpoints; c) meansfor determining whether each timing endpoint fails a threshold test; d)means for prioritizing any timing endpoints that fail the threshold testaccording to respective failure coefficients; and e) means for modifyingthe design to improve a slack of at least one of the timing endpoints.

A third aspect of the invention is directed to a computer programproduct comprising a computer useable medium having computer readableprogram code embodied therein for improving a probability of anintegrated circuit (IC) design meeting timing requirements, the programproduct comprising: a) program code configured to determine a referenceslack and a sensitivity of slack to a variation in at least oneparameter for each of a plurality of timing endpoints of the design; b)program code configured to calculate a failure coefficient from thereference slack and the sensitivity of slack for each of the timingendpoints; c) program code configured to determine whether each timingendpoint fails a threshold test; d) program code configured toprioritize any timing endpoints that fail the threshold test accordingto respective failure coefficients; and e) program code configured tomodify the design to improve a slack of at least one of the timingendpoints.

The foregoing and other features of the invention will be apparent fromthe following more particular description of embodiments of theinvention.

BRIEF DESCRIPTION OF THE DRAWINGS

The embodiments of this invention will be described in detail, withreference to the following figures, wherein like designations denotelike elements, and wherein:

FIG. 1 shows a conventional approach to timing analysis.

FIG. 2 shows a block diagram of an IC design improvement systemaccording to one embodiment of the invention.

FIG. 3 shows a flow diagram of operational methodology of the system ofFIG. 2 according to one embodiment of the invention.

FIG. 4 shows a flow diagram of processing by a sensitivity determinatorof FIG. 2.

FIG. 5 shows a flow diagram of steps S102 and S103 from the flow diagramof FIG. 3.

DETAILED DESCRIPTION OF THE INVENTION

For purposes of clarity only, the description includes the followingsub-titles: 1. System Overview, II. Operational Methodology, and Ill.Conclusion.

I. System Overview

With reference to the accompanying drawings, FIG. 2 is a block diagramof an integrated circuit (IC) design improvement system 100 according toone embodiment of the invention. System 100 includes a memory 112, aprocessing unit (PU) 114, input/output devices (I/O) 116 and a bus 118.A database 120 may also be provided for storage of data relative toprocessing tasks. Memory 112 includes a program product 122 that, whenexecuted by PU 114, comprises various functional capabilities describedin further detail below. Memory 112 (and database 120) may comprise anyknown type of data storage system and/or transmission media, includingmagnetic media, optical media, random access memory (RAM), read onlymemory (ROM), a data object, etc. Moreover, memory 112 (and database120) may reside at a single physical location comprising one or moretypes of data storage, or be distributed across a plurality of physicalsystems. PU 114 may likewise comprise a single processing unit, or aplurality of processing units distributed across one or more locations.I/O 116 may comprise any known type of input/output device including anetwork system, modem, keyboard, mouse, scanner, voice recognitionsystem, CRT, printer, disc drives, etc. Additional components, such ascache memory, communication systems, system software, etc., may also beincorporated into system 100. System 100 receives an IC design 200 to belegalized and outputs an improved IC design 202. It should be recognizedthat system 100 may be incorporated as a part of a larger IC designsystem or be provided as a separate system.

As shown in FIG. 2, program product 122 may include a sensitivitydeterminator 124, a distribution determinator 125, a failure coefficientcalculator 126, a threshold tester 128, a prioritizer 130, a modifier132, and other system components 134. Sensitivity determinator 124 mayinclude a slack determinator 140 and a sensitivity calculator 142.Failure coefficient calculator 126 may include a failure probability(FP) calculator 144 and a worst-case slack determinator 146. Othersystem components 134 may include any other necessary functionality notexpressly described herein.

It should be recognized that while system 100 has been illustrated as astandalone system, it may be included as part of a larger IC designsystem or a peripheral thereto. An IC design 200 is input to system 100,and an improved IC design 202 is output from system 100.

II. Operational Methodology

Turning to FIG. 3 in conjunction with FIG. 2, an operational methodologyof system 100 for improving a probability of an integrated circuit (IC)design meeting timing requirements according to one embodiment of theinvention will now be described. As a brief overview, the methodincludes the following steps: S101, determine a reference slack and asensitivity of slack to a variation in at least one parameter for eachof a plurality of timing endpoints of the design; S102, calculate afailure coefficient from the reference slack and the sensitivity ofslack for each of the timing endpoints; S103, determine whether at leastone timing endpoint has greater than a specified probability of failingat least one threshold test; S104, prioritize any timing endpoints thatfail the at least one threshold test according to their respectivefailure coefficients; and S105, modify the design to improve a slack ofat least one of the timing endpoints.

Turning to the details of the method, in first step S101, a referenceslack and a sensitivity of slack to a variation in at least oneparameter for each of a plurality of timing endpoints of the design aredetermined by sensitivity determinator 124. As used herein, a“parameter” can be one or more factors that can be varied to change thetiming of IC design 200. For example, a parameter may be a voltage, astructural (metal) dimension, a material characteristic, a process stepmeasurement, etc., or a combination of the above. With regard to thereference slack determination, slack determinator 140 may include anynow known or later developed software, along with hardware, necessary toobtain a slack value for a timing endpoint of IC design 200 (FIG. 2).

Referring to FIG. 4, a flow diagram of operational methodology ofsensitivity calculator 142 according to one embodiment of the inventionis shown. In this embodiment, in a first sub-step S101A, a firstreference slack Sj(ref1) is determined by sensitivity determinator 124for each timing endpoint by using reference values for all parameters,where j ranges between 1 and the total number of endpoints of interestdenoted as M. Each reference slack Sj(ref1) is calculated when all Nindependent parameters Xi, where i ranges between 1 and N, are set totheir corresponding reference parameter value Xi(ref1). Each referenceparameter value Xi(ref1) may be equivalent to, for example, an extremevalue from the range (distribution) of the parameter, a nominal valueXi(nom), or some other value. In a second sub-step S101B, sensitivitycalculator 142 determines a second slack Sj(Xi) for all M timingendpoints of interest when one parameter, Xi, is substantiallyequivalent to a new, different parameter value Xi(2). In one embodiment,the different parameter value Xi(2) may be a value corresponding to anopposite, second extreme value from the range (distribution) of theparameter that was chosen for the reference parameter value during thereference slack calculation step S101A. During this step, only parameterXi is changed while all other N-1 parameter values remain set to thevalues used in sub-step S101A. In one embodiment, the value for areference parameter used during sub-step S101A is Xi(nom)+Ki*σi and thevalue used for the different parameter value during sub-step S101B isXi(nom)−Ki*σi, where Xi(nom) is the mean, σi is the standard deviation,and Ki is the standard deviation multiplier of the Xi parameterdistribution. Typically, a value of 3 is chosen for Ki to represent ±3standard deviations from the mean, however, other values may be used. Insub-step S101C, sensitivity calculator 142 calculates the slacksensitivity (∂Sj/∂Xi) to parameter Xi variations for all M timingendpoints of interest. In one embodiment, the slack sensitivity for agiven timing endpoint is calculated by dividing the difference in slack∂Sj(i.e., Sj(ref1)−Sj(Xi)) by the difference in the parameter's value∂Xi(i.e., Xi(ref1)−Xi(2)) in order to obtain ∂Sj/∂Xi, however othermethods of calculating sensitivity of slack to parameter variation maybe used. It should be recognized that the two slack computations (stepS101A and S101B) used to determine each slack sensitivity ∂Sj/∂Xi(stepS101C) may give different results due to different paths to endpoint j.Thus, they are not simply delay derivatives, but include effects due tothe dominance of different paths under different parameter/processpermutations. Once the slack sensitivities for all M timing endpointsare calculated for parameter Xi, steps S101B and S101C are repeated foreach of the remaining N-1 parameters.

Returning to FIG. 3, the second step S102 includes failure coefficientcalculator 144 calculating a failure coefficient (FP) from the referenceslack Sj(ref1) and parameter sensitivities, i.e., ∂Sj/∂XI to ∂Sj/∂XN,for each of the timing endpoints. As used herein, a “failurecoefficient” is a value indicative of a probability of failure for atiming endpoint, and can be embodied in a number of ways includingactual numerical calculations of a failure probability and a worst caseexcess slack, as will be described below. In step S103, a determinationis made by threshold tester 128 as to whether each timing endpoint failsa threshold test. As will become clear below, the failure coefficientcalculated and the threshold test evaluated, and the order of step S102and S103, depends on what data is available based on the parameterdistributions obtained as part of step S101.

Turning to FIG. 5, details of steps S102 and S103 will now be described.

In a first sub-step S201, a worst-case (WC) slack determinator 146calculates worst case (minimum or least positive) slack using theendpoint's reference slack Sj(ref1) and parameter sensitivities ∂Sj/∂XIto ∂Sj/∂XN. In one embodiment, a worst case (minimal) slack value ascomputed by WC slack determinator 146 as: S(Yσ)=Sj(nom)−Y*Sum over all Nparameters of ABS {∂Sj/∂Xi*σi}, where ABS is the absolute value and Y isa user definable multiple. The nominal slack for the slack distributioncan be calculated as: Sj(nom)=Sj(ref1)−sum over all N parameters{∂Sj/∂Xi*Ki*σi}, where Ki*σi is the difference between values Xi(nom)and Xi(ref) of parameter i. The worst case slack S(Yσ) is computed asthe difference between nominal slack and a multiple (Y) of a sum overall parameters of an absolute value of standard deviation of the delayvariation with respect to that parameter. In an alternative embodiment,the worst-case slack for each timing endpoint can be calculated as adifference between a nominal slack and a multiple (Y) of a root sumsquare over all parameters of an absolute value of standard deviation.

Next in sub-step S202, a determination is made as to whether allparameter distributions are defined in such a way that a failureprobability can be determined (step S205) by distribution determinator125. This determination may include determining whether thedistributions are Gaussian or can be readily modeled as Gaussian. If YESat sub-step S202, then processing proceeds to sub-steps S203-S204,otherwise, processing proceeds to sub-steps S205-S206.

1. Sub-Steps S203-S204: Parameter Distributions Known: Gaussian orModeled as Gaussian

If all parameter distributions are Gaussian or modeled as Gaussian, afailure coefficient can be calculated as a failure probability by FPcalculator at sub-step S203 in one of three ways:

First, the failure coefficient can be calculated as a failureprobability by FP calculator 144 based on a user-specified multiple (Y)of a standard deviation (sigma or σ) of the slack distribution, i.e.,the ±Y sigma points of the slack distribution. This step may becompleted by first determining the nominal slack for the slackdistribution as: Sj(nom)=Sj(ref1)−sum over all N parameters{∂Sj/∂Xi*Ki*σi}, where Ki*σi is the difference between values Xi(nom)and Xi(ref) of parameter i. The failure coefficient S(Yσ) can then becomputed by determining an endpoint slack at a specified Y sigmamultiplier as: S(Yσ)=Sj(nom)−Y*RSS over all N parameters {∂Sj/∂Xi*σi},where RSS is the root sum square. Typically, Y will be 3, but othersigma multipliers could be used.

Second, the failure probability can be calculated by FP calculator 144by determining a sigma multiplier Z that would result in the slack ofthat endpoint equaling a specified slack threshold S(threshold).Specifically, Z would be computed as: Z=(Sj(nom)−S(threshold))/(RSS overall N parameters {∂Sj/∂Xi*σi}), where RSS is the root sum square.Sj(nom) would be calculated as described above. A larger value of Zwould indicate a larger probability of failure.

Third, a failure probability could be calculated by FP calculator 144 byintegrating the slack probability distribution from minus infinity (−∞)to Sj(nom)−Z*RSS over all N parameters {∂Sj/∂Xi*σi} using an errorfunction of a Gaussian density (i.e., an erf function). The sigmamultiplier Z would be calculated as described above.

Once the failure probability is known, at sub-step S204, a determinationis made by threshold tester 128 as to whether a failure probability ofany timing endpoint exceeds a failure coefficient threshold R. Ingeneral, a failure coefficient threshold R may be selected by a user tobalance performance for risk of loss of product yield. If NO at sub-stepS204, then no IC design modification is necessary and processing ends.Otherwise, i.e., YES at sub-step S204, processing returns to step S104of FIG. 3.

2. Sub-Steps S205-S206: Parameter Distributions Undefined

If the distributions are undefined, i.e., NO at sub-step S202, insub-step S205, a determination is made by threshold tester 128 as towhether a worst-case WC slack S(Yσ) (computed for each timing endpointby worst case slack determinator 146 (sub-step S201)) of any timingendpoint is less positive than a slack threshold S(threshold). Fortypical timing applications, slack threshold is usually zero, but thisis not necessary. If any worst-case slack S(Yσ) is more positive thanslack threshold S(threshold), i.e., NO at sub-step S205, the probabilityof failure is considered to be zero and no IC design modification isnecessary, so processing ends. That is, timing endpoints having aworst-case WC slack S(Yσ) value greater than some user-specifiedthreshold S(threshold) (typically zero) are considered to have a zerofailure probability and thus need no improvement. If any worst-caseslack S(Yσ) is more negative than slack threshold S(threshold), i.e.,YES at sub-step S205, the probability of failure is non-zero andprocessing proceeds to sub-step S206. Among timing endpoints with worstcase WC slack S(Yσ) values greater than S(threshold), those with lower(e.g., more negative) values are considered to have higher failureprobabilities and thus have a higher priority for improvement.

In sub-step S206, the worst case (WC) slack values S(Yσ) computed foreach timing endpoint by worst case slack determinator 146 (sub-stepS201) are used by failure coefficient calculator 126 to generate afailure coefficient in the form of a worst case slack excess, whichequals S(threshold) −S(Yσ). That is, the probability of failure for atiming endpoint can be derived from the worst case (minimum or leastpositive) slack value.

With regard to sub-steps S203 and S206, it should be recognized that thetypes of thresholds can vary depending on how the failure coefficient iscalculated. In particular, since Z (the sigma multiple at which theslack becomes less than a slack threshold), S(Yσ) (the slack at aparticular sigma multiple−computed by either RSS or summing theindividual parameter sigma contributions) and an actual failureprobability (computed by integrating a Gaussian distribution), aredifferent numbers with different dimensions and ranges, the thresholdsagainst which each should be compared may be different. For example, inthe event that a sigma multiplier Z (at which a specified slackthreshold S(threshold) is reached) is used as a failure coefficient,S(threshold) will be a sigma multiplier against which Z is compared.Accordingly, the failure coefficient used may be user selected, and anappropriate failure coefficient threshold used.

The results from sub-steps S203 or S206 are failure coefficients foreach of the M timing endpoints. With further regard to sub-steps S203and S206, it is also possible to introduce multiple pairs of slackthresholds S(threshold_i) and failure coefficient thresholds Ri, so thatsystem 100 could analyze using stepped slack thresholds S(threshold_i)and failure coefficient thresholds Ri. In this fashion, modificationrequirement decisions can be made based on the various points of theslack distribution, rather than on just a single point.

Returning to FIG. 3, in fourth step S104, any timing endpoints that failat least one threshold test are prioritized by prioritizer 130 accordingto their respective failure coefficients, i.e., failure probability orworst case slack excess. In one embodiment, the timing endpoints areprioritized from a highest failure coefficient (i.e., those most likelyrequiring modification) to a lowest failure coefficient (i.e., thoseless likely to require modification). For example, a higher prioritywould be assigned to endpoints with higher WC slack excessesS(threshold)−S(Yσ), lower computed Z sigma multipliers at which aspecified slack threshold is reached, or at higher actual computedfailure probabilities. Where multiple pairs of slack/failure coefficientthresholds (S(threshold_i)/Ri) are used, the prioritization may also bebased on the multiplicity of threshold testing pairs. For example,system 100 may choose to modify all timing endpoints whose WC slack isless than slack threshold S(threshold_0) with a failure coefficientgreater than R0, and then modify remaining timing endpoints whose WCslack is less than slack threshold X S(threshold_1) with a failurecoefficient greater than R1, and so on. The prioritization may alsoinclude an estimation of the cost to implement a modification, e.g.,increase in area, power, wiring required, etc., and implementmodifications first for timing endpoints where the ratio between thefailure coefficient and the cost of the modification is a maximum.

Next in step S105, the design is modified by modifier 132 to improve theslack of at least one of the timing endpoints. An improved IC design 202is output. Modifier 132 may include any now known or later developedsoftware mechanisms for correcting IC design 200 to address the timingissues. Where the prioritization is from highest to lowest failurecoefficient, the modifying step includes modifying the design byaddressing the timing endpoints from a failure coefficient indicative ofa highest probability of failure to a failure coefficient indicative ofa lowest probability of failure. Timing margin is traditionally added tothe endpoint by padding either the early or late path delay thatconverges at the endpoint. Modifier 132 may also make modificationsusing parameter sensitivity for a respective timing endpoint byproviding a sufficient timing margin for a respective timing endpoint inorder to account for sensitivities to parameter variations. In addition,a high value for ∂Sj/∂Xi indicates that timing endpoint Sj has a highsensitivity to variations in parameter Xi. This implies a largeimbalance of parameter Xi between the early and late timing paths. Tocorrect for this imbalance, the sensitivity to parameter variations fora respective endpoint are used to guide parameter balancing andconnectivity between paths that compose a timing test at the timingendpoint. That is, the layout between late and early paths are balancedor matched with similar parameter content and connectivity. For example,if a late path is routed primarily on metal layer C and the early pathis routed primary on metal layer D, the timing endpoint associated withthese paths will exhibit sensitivity to both metal layers C and D. Themagnitude of the sensitivity is proportional to the amount of wire ineach path. To reduce this sensitivity, one approach is to balance themetal layer content and connectivity between the early and lath paths inorder to facilitate path delay tracking, thereby reducing slacksensitivity to metal C and metal D variations. Once modifications aremade to the design to correct for timing violations, either by delaypadding or by parameter-connectivity matching between the early and latepaths, the process is repeated by returning to step S101 in FIG. 3. Ifall modifications in step S105 were successful, the new failurecoefficients for each timing endpoint calculated from substeps S203(FIG. 5) and S206 (FIG. 5) will terminate the process. If additionalfixes are required, the processing flow of FIG. 5 is repeated.

III. Conclusion

The above-described invention provides a way, based on the knownsensitivities, to provide a number of advantageous functions. Forexample, when parameters are independent from each other the inventionallows one to determine what point within the 2^(N) parameter spacecorresponds to the most conservative physical space. This allows one toquickly identify the process space for timing closure on each endpointwithout doing 2^(N) timing runs. In addition, the invention allows usingthe sensitivities to calculate a slack adjustment on the referencetiming run that would correspond to the slack that would be obtained atthe most conservative physical process space. The sensitivities can alsobe used to guide modification methodologies. Parameters with the highestsensitivities illustrate an unbalanced content of parameter usagebetween the early and late paths, which can be used to directcorrection. The probability of endpoint failures can be used toprioritize timing endpoints for modification or help guide hardwarefunctional test on which endpoints to test.

In the previous discussion, it will be understood that the method stepsdiscussed are performed by a processor, such as PU 114 of system 100,executing instructions of program product 122 stored in memory. It isunderstood that the various devices, modules, mechanisms and systemsdescribed herein may be realized in hardware, software, or a combinationof hardware and software, and may be compartmentalized other than asshown. They may be implemented by any type of computer system or otherapparatus adapted for carrying out the methods described herein. Atypical combination of hardware and software could be a general-purposecomputer system with a computer program that, when loaded and executed,controls the computer system such that it carries out the methodsdescribed herein. Alternatively, a specific use computer, containingspecialized hardware for carrying out one or more of the functionaltasks of the invention could be utilized. The present invention can alsobe embedded in a computer program product, which comprises all thefeatures enabling the implementation of the methods and functionsdescribed herein, and which−when loaded in a computer system—is able tocarry out these methods and functions. Computer program, softwareprogram, program, program product, or software, in the present contextmean any expression, in any language, code or notation, of a set ofinstructions intended to cause a system having an information processingcapability to perform a particular function either directly or after thefollowing: (a) conversion to another language, code or notation; and/or(b) reproduction in a different material form.

While this invention has been described in conjunction with the specificembodiments outlined above, it is evident that many alternatives,modifications and variations will be apparent to those skilled in theart. Accordingly, the embodiments of the invention as set forth aboveare intended to be illustrative, not limiting. Various changes may bemade without departing from the spirit and scope of the invention asdefined in the following claims.

1. A method of improving a probability of an integrated circuit (IC)design meeting timing requirements, the method comprising the steps of:a) determining a reference slack and a sensitivity of slack to avariation in at least one parameter for each of a plurality of timingendpoints of the design; b) calculating a failure coefficient from thereference slack and the sensitivity of slack for each of the timingendpoints; c) determining whether each timing endpoint fails a thresholdtest; d) prioritizing any timing endpoints that fail the threshold testaccording to respective failure coefficients; and e) modifying thedesign to improve a slack of at least one of the timing endpoints. 2.The method of claim 1, wherein the sensitivity determining stepincludes: determining a first slack at each timing endpoint for areference parameter value for each parameter; determining a second slackat each timing endpoint for a different parameter value than thereference parameter value for each parameter; and calculating thesensitivity of the slack to the variation in the at least one parameter.3. The method of claim 2, wherein the reference parameter value issubstantially equivalent to one of: a first extreme value of a valuerange of the parameter; and a nominal value plus a multiple of astandard deviation of the parameter.
 4. The method of claim 3, whereinin the case that the first extreme value is used, the differentparameter value is substantially equivalent to an opposite, secondextreme value of the value range of the parameter.
 5. The method ofclaim 3, wherein the sensitivity calculating step includes dividing adifference between the first slack and the second slack by a differencebetween the reference parameter value and the different parameter value.6. The method of claim 1, wherein the failure coefficient calculatingand the threshold test determining steps include: calculating a worstcase slack for each timing endpoint; and determining whether a parameterdistribution is Gaussian or non-Gaussian.
 7. The method of claim 6,wherein the worst case slack calculating step includes one of:determining a difference between a nominal slack and a multiple (Y) of asum over all parameters of an absolute value of standard deviation; anddetermining a difference between a nominal slack and a multiple (Y) of aroot sum square over all parameters of an absolute value of standarddeviation.
 8. The method of claim 6, wherein in the case that theparameter distribution is Gaussian, the failure coefficient calculatingstep includes calculating the failure coefficient for each timingendpoint by one of: a) integrating a slack distribution from minusinfinity to a failure threshold; b) determining a sigma multiplier atwhich an endpoint slack equals a slack threshold; and c) calculating amultiple of a standard deviation of a slack distribution; and thethreshold test includes the failure probability exceeding a failureprobability threshold.
 9. The method of claim 6, wherein in the casethat the parameter distribution is non-Gaussian, the threshold testincludes the worst-case slack of any timing endpoint exceeding a slackthreshold, and the failure coefficient calculating step includescalculating a worst case slack excess for any timing endpoint exceedingthe slack threshold.
 10. The method of claim 1, wherein the modifyingstep includes modifying the design by addressing the timing endpointsfrom a failure coefficient indicative of a highest probability offailure to a failure coefficient indicative of a lowest probability offailure.
 11. The method of claim 1, wherein the modifying step includesusing the sensitivity to parameter variations for a respective timingendpoint to guide parameter balancing and connectivity between pathsthat compose a timing test at the timing endpoint.
 12. The method ofclaim 1, wherein the modifying step includes providing a sufficienttiming margin for a respective timing endpoint in order to account forsensitivities to parameter variations.
 13. A system for improving aprobability of an integrated circuit (IC) design meeting timingrequirements, the system comprising: a) means for determining areference slack and a sensitivity of slack to a variation in at leastone parameter for each of a plurality of timing endpoints of the design;b) means for calculating a failure coefficient from the reference slackand the sensitivity of slack for each of the timing endpoints; c) meansfor determining whether each timing endpoint fails a threshold test; d)means for prioritizing any timing endpoints that fail the threshold testaccording to respective failure coefficients; and e) means for modifyingthe design to improve a slack of at least one of the timing endpoints.14. The system of claim 13, wherein the sensitivity determining meansincludes means for: determining a first slack at each timing endpointfor a reference parameter value for each parameter; determining a secondslack at each timing endpoint for a different parameter value than thereference parameter value for each parameter; and calculating thesensitivity of the slack to the variation in the at least one parameter.15. The system of claim 14, wherein the reference parameter value issubstantially equivalent to one of: a first extreme value of a valuerange of the parameter; and a nominal value plus a multiple of astandard deviation of the parameter.
 16. The system of claim 14, whereinin the case that the first extreme value is used, the differentparameter value is substantially equivalent to an opposite, secondextreme value of the value range of the parameter.
 17. The system ofclaim 14, wherein the sensitivity calculating means includes means fordividing a difference between the first slack and the second slack by adifference between the reference parameter value and the differentparameter value.
 18. The system of claim 13, wherein the failurecoefficient calculating means and the threshold test determining meansinclude means for: calculating a worst case slack for each timingendpoint; and determining whether a parameter distribution is Gaussianor non-Gaussian.
 19. The system of claim 18, wherein the worst caseslack calculating means calculates the worst case slack by one of:determining a difference between a nominal slack and a multiple (Y) of asum over all parameters of an absolute value of standard deviation; anddetermining a difference between a nominal slack and a multiple (Y) of aroot sum square over all parameters of an absolute value of standarddeviation.
 20. The system of claim 18, wherein in the case that theparameter distribution is Gaussian, the failure coefficient calculatingmeans calculates the failure coefficient for each timing endpoint by oneof: a) integrating a slack distribution from minus infinity to a failurethreshold; b) determining a sigma multiplier at which an endpoint slackequals a slack threshold; and c) calculating a multiple of a standarddeviation of a slack distribution; and the threshold test includes thefailure probability exceeding a failure probability threshold.
 21. Thesystem of claim 18, wherein in the case that the parameter distributionis non-Gaussian, the threshold test includes the worst-case slack of anytiming endpoint exceeding a slack threshold, and the failure coefficientcalculating means calculates a worst case slack excess for any timingendpoint exceeding the slack threshold.
 22. The system of claim 13,wherein the modifying means modifies the design by addressing the timingendpoints from a failure coefficient indicative of a highest probabilityof failure to a failure coefficient indicative of a lowest probabilityof failure.
 23. The system of claim 13, wherein the modifying means usesthe sensitivity to parameter variations for a respective timing endpointto guide parameter balancing and connectivity between paths that composea timing test at the timing endpoint.
 24. The system of claim 13,wherein the modifying means provides a sufficient timing margin for arespective timing endpoint in order to account for sensitivities toparameter variations.
 25. A computer program product comprising acomputer useable medium having computer readable program code embodiedtherein for improving a probability of an integrated circuit (IC) designmeeting timing requirements, the program product comprising: a) programcode configured to determine a reference slack and a sensitivity ofslack to a variation in at least one parameter for each of a pluralityof timing endpoints of the design; b) program code configured tocalculate a failure coefficient from the reference slack and thesensitivity of slack for each of the timing endpoints; c) program codeconfigured to determine whether each timing endpoint fails a thresholdtest; d) program code configured to prioritize any timing endpoints thatfail the threshold test according to respective failure coefficients;and e) program code configured to modify the design to improve a slackof at least one of the timing endpoints.
 26. The program product ofclaim 25, wherein the sensitivity determining code includes program codeconfigured to: determine a first slack at each timing endpoint for areference parameter value for each parameter; determine a second slackat each timing endpoint for a different parameter value than thereference parameter value for each parameter; and calculate thesensitivity of the slack to the variation in the at least one parameter.27. The program product of claim 26, wherein the reference parametervalue is substantially equivalent to one of: a first extreme value of avalue range of the parameter; and a nominal value plus a multiple of astandard deviation of the parameter.
 28. The program product of claim26, wherein in the case that the first extreme value is used, thedifferent parameter value is substantially equivalent to an opposite,second extreme value of the value range of the parameter.
 29. Theprogram product of claim 26, wherein the sensitivity calculating codeincludes program code configured to divide a difference between thefirst slack and the second slack by a difference between the referenceparameter value and the different parameter value.
 30. The programproduct of claim 25, wherein the failure coefficient calculating codeand the threshold test determining code include program code configuredto: calculate a worst case slack for each timing endpoint; and determinewhether a parameter distribution is Gaussian or non-Gaussian.
 31. Theprogram product of claim 30, wherein the worst case slack calculatingcode calculates the worst case slack by one of: determining a differencebetween a nominal slack and a multiple (Y) of a sum over all parametersof an absolute value of standard deviation; and determining a differencebetween a nominal slack and a multiple (Y) of a root sum square over allparameters of an absolute value of standard deviation.
 32. The programproduct of claim 30, wherein in the case that the parameter distributionis Gaussian, the failure coefficient calculating code calculates thefailure coefficient for each timing endpoint by one of: a) integrating aslack distribution from minus infinity to a failure threshold; b)determining a sigma multiplier at which an endpoint slack equals a slackthreshold; and c) calculating a multiple of a standard deviation of aslack distribution; and the threshold test includes the failureprobability exceeding a failure probability threshold.
 33. The programproduct of claim 30, wherein in the case that the parameter distributionis non-Gaussian, the threshold test includes the worst-case slack of anytiming endpoint exceeding a slack threshold, and the failure coefficientcalculating code calculates a worst case slack excess for any timingendpoint exceeding the slack threshold.
 34. The program product of claim25, wherein the modifying code modifies the design by addressing thetiming endpoints from a failure coefficient indicative of a highestprobability of failure to a failure coefficient indicative of a lowestprobability of failure.
 35. The program product of claim 25, wherein themodifying code uses the sensitivity to parameter variations for arespective timing endpoint to guide parameter balancing and connectivitybetween paths that compose a timing test at the timing endpoint.
 36. Theprogram product of claim 25, wherein the modifying code provides asufficient timing margin for a respective timing endpoint in order toaccount for sensitivities to parameter variations.